Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in JAVA):

Solution:
import java.io.OutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.Arrays;
import java.util.Scanner;

/**
 * Built using CHelper plug-in
 * Actual solution is at the top
 *
 * @author silviase
 */
public class Main {
    public static void main(String[] args) {
        InputStream inputStream = System.in;
        OutputStream outputStream = System.out;
        Scanner in = new Scanner(inputStream);
        PrintWriter out = new PrintWriter(outputStream);
        ICoins solver = new ICoins();
        solver.solve(1, in, out);
        out.close();
    }

    static class ICoins {
        public void solve(int testNumber, Scanner in, PrintWriter out) {
            int n = in.nextInt();
            double[] dp = new double[n + 1];
            Arrays.fill(dp, 0);
            dp[0] = 1;

            for (int i = 0; i < n; i++) {
                double p = in.nextDouble();
                for (int j = dp.length - 1; j >= 0; j--) {
                    if (j > 0) {
                        dp[j] = dp[j] * (1 - p) + dp[j - 1] * p;
                    } else {
                        dp[0] *= (1 - p);
                    }
                }
            }

            double res = 0;
            for (int i = (n + 1) / 2; i <= n; i++) {
                res += dp[i];
            }
            out.println(res);

        }

    }
}